Analysing the Moments of the Determinant of a Random Matrix Via Analytic Combinatorics of Permutation Tables
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| Format: | Preprint |
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2025
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| _version_ | 1866912465947721728 |
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| author | Beck, Dominik Lv, Zelin Potechin, Aaron |
| author_facet | Beck, Dominik Lv, Zelin Potechin, Aaron |
| contents | We consider the following natural question. Given a matrix $A$ with i.i.d. random entries, what are the moments of the determinant of $A$? In other words, what is $\mathbb{E}[\det(A)^k]$? While there is a general expression for $\mathbb{E}[\det(A)^k]$ when the entries of $A$ are Gaussian, much less is known when the entries of $A$ have some other distribution.
In two recent papers, we answered this question for $k = 4$ when the entries of $A$ are drawn from an arbitrary distribution and for $k = 6$ when the entries of $A$ are drawn from a distribution which has mean $0$. These analyses used recurrence relations and were highly intricate. In this paper, we show how these analyses can be simplified considerably by using analytic combinatorics on permutation tables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03651 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysing the Moments of the Determinant of a Random Matrix Via Analytic Combinatorics of Permutation Tables Beck, Dominik Lv, Zelin Potechin, Aaron Combinatorics Probability 05A05, 05A15 We consider the following natural question. Given a matrix $A$ with i.i.d. random entries, what are the moments of the determinant of $A$? In other words, what is $\mathbb{E}[\det(A)^k]$? While there is a general expression for $\mathbb{E}[\det(A)^k]$ when the entries of $A$ are Gaussian, much less is known when the entries of $A$ have some other distribution. In two recent papers, we answered this question for $k = 4$ when the entries of $A$ are drawn from an arbitrary distribution and for $k = 6$ when the entries of $A$ are drawn from a distribution which has mean $0$. These analyses used recurrence relations and were highly intricate. In this paper, we show how these analyses can be simplified considerably by using analytic combinatorics on permutation tables. |
| title | Analysing the Moments of the Determinant of a Random Matrix Via Analytic Combinatorics of Permutation Tables |
| topic | Combinatorics Probability 05A05, 05A15 |
| url | https://arxiv.org/abs/2507.03651 |